st: Constructing confidence intervals for a sum of forecasts

I am generating a dynamic forecast of two variables, x and y, after
running a vector autoregression, and I want to construct the forecast
confidence interval for the sum of their first differences, D.x + D.y.
My question is what is the correct way to do this.

I can easily generate the se's of the first differences of individual
variables using -fcast compute, diff-. Let's call these se_Dx_hat and
se_Dy_hat. It is simple to compute the se of their sum if I assume
independence:

se(Dx_hat+Dy_hat) = sqrt( se_Dx_hat^2 + se_Dy_hat^2 )

However, I am not clear about how to correctly handle potential
correlation in the first-differences, which this expression omits.

Should I just assume that each of the terms in square brackets is a
constant, given by, first, the data, and second, the appropriate
elements in e(V) generated by my var? If not, can anyone recommend an
alternative way of doing this calculation?